A solution of the Ten Martini Problem (Q2577055)

The author reports on the research concerning the conjecture (known as the ``Ten Martini Problem'' after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all nonzero values of the coupling and all irrational frequencies. Partial results were obtained, among others, by \textit{M.--D.\ Choi, G.\ A.\ Elliott} and \textit{N.\ Yui} [Invent. Math. 99, No. 2, 225--246 (1990; Zbl 0665.46051)] and, using a combination of techniques from spectral and dynamical systems theory, by the author [Commun. Math. Phys. 244, No. 2, 297--309 (2004; Zbl 1075.39021)]. With the same approach, a complete proof has recently been achieved and submitted by \textit{A.\ Avila} and \textit{S.\ Jitomirskaya} [``The Ten Martini Problem'' (arXiv preprint) (2005; arXiv:math.DS/0503363)].